The book mentioned in a comment, Sequences of Real Numbers, by Sîntămărian & Furdui, presents on page 11 the recursion $$x_{n+1}=x_n+\frac{n^{2\beta}}{x_1+x_2+\cdots+x_n}$$ and the derivation of the limit $$\lim_{n\rightarrow\infty}x_n/n^\beta=\sqrt{1+1/\beta}$$ as an application of the Stolz–Cesàro Theorem.
The OP asks where this recursion relations might appear in a research context. Consider the integrodifferential equation $$\left(\int_0^t f(t')dt'\right)\frac{d}{dt}f(t)=(t/\tau)^{2\beta},\;\;\tau>0.$$ The discretization $t\mapsto n\tau$, $n=0,1,2\ldots$, with $f_n=f(n\tau)$ would then read $$f_{n+1}-f_n=\frac{n^{2\beta}}{\sum_{k=1}^n f_k},$$ which is the recursion relation under investigation.